The primary goal of seismic exploration is to obtain images of subsurface geological formations based upon information gleaned from recordings of a class of acoustic wave signals usually referred to as a seismic wavefield which is purposefully directed into the earth and recorded after having traveled through sections of the earth.
The propagation of the seismic wavefield from the source to the receivers is dominated by geophysical properties of the layers through which the signals travel. The interpretation of the seismic wavefield yields in the first instance acoustic properties of the layers, the most important of which are known as elastic constants, which in turn can be used to determine other properties such as rock type, composition and fluid content, or texture and porosity.
Seismic processing usually involves evaluating the data to identify events and their traveltimes. By matching the detected traveltimes of events and the location of the receivers, velocity information is gained from the data and signals recorded by different receivers can be stacked to increase the signal-to-noise ratio of the recorded data. In a subsequent step, often referred to as imaging or migration, the data is migrated from its position in time to its equivalent position in depth or, more generally, its position in the subsurface. As a result of this seismic processing, an image of the subsurface is generated to be used for stratigraphic and geological interpretation.
Apart from the event- and traveltime-based methods, theoretical and experimental methods have been proposed that attempt an inversion based on the energy content of the recorded signal rather than just the location and time of identified events. These methods are commonly referred to as full wavefield or waveform inversion.
At the most basic level, seismic waveform inversion methods can be classified as either direct or iterative, the latter being more common. The iterative methods need a starting model of the earth. This is usually in the form of a velocity-versus-depth model which must not be too far removed from the true structure. The starting model is modified using a gradient-descent method in order to fit waveforms in a least-squares sense. In 3D problems, the calculations for an iterative process require significant computational resources. An interesting common feature of the iterative results is that during the early stages of data fitting the models are in fact rather smooth and relatively simple.
Direct inversion, by contrast, involves no starting model, no gradient or Fréchet-derivative evaluation and obtains the velocity in one step. The theory of direct inversion is well-established for backscattered waves in a plane layered medium. In fact, for scalar waves in a stratified medium, a single horizontal slowness or angle of incidence suffices to find the velocity structure from the times and amplitudes of reflections. Having more than one incident lateral slowness provides redundancy.
Apart from waves reflected at impedance steps within a subterranean region, there are waves the path of which is reversed from downgoing to upgoing by a gradual velocity change rather than by encountering a sharp reflector. Members of this subset of the recorded wavefield are called “turning waves” or “diving waves”.
For turning waves, the only comparable direct inversion result to date has been the famous Herglotz-Wiechert-Bateman explicit formula converting their measured arrival times to the velocity-depth profile outside low-velocity zones, as described in standard textbooks, for example K. Aki & P. G. Richards, “Quantitative Seismology”, University Science Books, Sausalitp, Ca., USA, 414-429.
Furthermore, turning waves have been used in migration or imaging of overhanging salt, for example in U.S. Pat. Nos. 5,138,584; 5,235,555 and 5,490,120. These known methods are imaging methods based essentially on phase or traveltime analysis, as in the Herglotz-Wiechert-Bateman formula.
R. W. Clayton and George A McMechan in: “Inversion of refraction data by wave field continuation”, Geophysics Vol. 46. No 6 (June 1981), pp. 860-868, described a method of inverting refraction data using downward continuation in a specified background model and iterating to adjust that model. This is analogous to the Herglotz-Wiechert-Bateman traveltime method and is a phase, extrapolation method which does not use the amplitudes or energies. This method does not make use of true amplitude extrapolation. It is also an iterative method relying an initial guess or estimate of a starting velocity model. The authors required ad hoc phase shifts, e.g. 5π/4, to obtain reasonable results.